Hydrodynamically Coupled Brownian Dynamics: A coarse-grain particle-basedBrownian dynamics technique with hydrodynamic interactions for modeling self-developing flow of polymer solutionsV. R. Ahuja, J. van der Gucht, and W. J. Briels

Citation: The Journal of Chemical Physics 148, 034902 (2018); doi: 10.1063/1.5006627View online: https://doi.org/10.1063/1.5006627View Table of Contents: http://aip.scitation.org/toc/jcp/148/3Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 148, 034902 (2018)

Hydrodynamically Coupled Brownian Dynamics: A coarse-grainparticle-based Brownian dynamics technique with hydrodynamicinteractions for modeling self-developing flow of polymer solutions

V. R. Ahuja,1,2,3,a) J. van der Gucht,3 and W. J. Briels1,2,4,b)1Computational Chemical Physics, Faculty of Science and Technology, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands2MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede,The Netherlands3Physical Chemistry and Soft Matter, Wageningen University, Building 124, Stippeneng 4,6708 WE, Wageningen, The Netherlands4Forschungszentrum Julich, ICS 3, D-52425 Julich, Germany

(Received 26 September 2017; accepted 21 December 2017; published online 16 January 2018)

We present a novel coarse-grain particle-based simulation technique for modeling self-developing flowof dilute and semi-dilute polymer solutions. The central idea in this paper is the two-way couplingbetween a mesoscopic polymer model and a phenomenological fluid model. As our polymer model, wechoose Responsive Particle Dynamics (RaPiD), a Brownian dynamics method, which formulates theso-called “conservative” and “transient” pair-potentials through which the polymers interact besidesexperiencing random forces in accordance with the fluctuation dissipation theorem. In addition tothese interactions, our polymer blobs are also influenced by the background solvent velocity field,which we calculate by solving the Navier-Stokes equation discretized on a moving grid of fluidblobs using the Smoothed Particle Hydrodynamics (SPH) technique. While the polymers experiencethis frictional force opposing their motion relative to the background flow field, our fluid blobs alsoin turn are influenced by the motion of the polymers through an interaction term. This makes ourtechnique a two-way coupling algorithm. We have constructed this interaction term in such a waythat momentum is conserved locally, thereby preserving long range hydrodynamics. Furthermore, wehave derived pairwise fluctuation terms for the velocities of the fluid blobs using the Fokker-Planckequation, which have been alternatively derived using the General Equation for the Non-EquilibriumReversible-Irreversible Coupling (GENERIC) approach in Smoothed Dissipative Particle Dynamics(SDPD) literature. These velocity fluctuations for the fluid may be incorporated into the velocityupdates for our fluid blobs to obtain a thermodynamically consistent distribution of velocities. Incases where these fluctuations are insignificant, however, these additional terms may well be droppedout as they are in a standard SPH simulation. We have applied our technique to study the rheologyof two different concentrations of our model linear polymer solutions. The results show that thepolymers and the fluid are coupled very well with each other, showing no lag between their velocities.Furthermore, our results show non-Newtonian shear thinning and the characteristic flattening ofthe Poiseuille flow profile typically observed for polymer solutions. Published by AIP Publishing.https://doi.org/10.1063/1.5006627

I. INTRODUCTION

Simulating the dynamics of polymers in solution has beena subject of continuous interest for almost three decades now,yet a final resolution of the problem eludes us owing to itsinherent complexity. This problem poses many fundamentalquestions such as how do these large polymer molecules inter-act with the solvent fluid molecules and how do they exchangemomentum under flowing conditions? Given the enormoussize of the polymer molecules dissolved in a fluid made upof molecules which are orders of magnitude smaller, it ispractically impossible to even simulate just tens of polymer

a)Electronic mail: v.r.ahuja@utwente.nlb)Electronic mail: w.j.briels@utwente.nl

molecules together with all the millions of fluid molecules intheir vicinity if all of them are to be resolved down to the atomiclevel. Hence, coarse-graining is an inherent part of simulatingpolymers in solution.

The obvious question that arises next is what should bethe level of coarse-graining, i.e., how well do we resolvethe polymer molecules and the fluid? Very broadly speak-ing, one can categorize the simulation techniques into twomajor branches—particle-based simulations and Computa-tional Fluid Dynamics (CFD) simulations using a continuumapproach. Since there is a plethora of techniques under eachcategory, we restrict the discussion here to particle-based sim-ulation techniques, which, we might add, have the advantageof being able to link macroscopic flow behavior to microscopicinteractions. Among the particle-based simulation techniques

0021-9606/2018/148(3)/034902/11/$30.00 148, 034902-1 Published by AIP Publishing.

034902-2 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

that have been used over the years to model the dynamics ofpolymers in solution, there is again an enormous variety ofmethods employing different levels of coarse-graining for thepolymers and the fluid. There have been studies employingmolecular dynamics (MD) for simulating polymers modeledas a chain of a few “monomers” connected to each other withsprings in a solvent represented by a few thousand fluid parti-cles treated as soft spheres.1,2 Hybrid models have been devel-oped which combine two different levels of coarse-grainingfor the polymers and the fluid. One such hybrid model com-bines the MD approach for the polymer chain with the Lat-tice Boltzmann (LB) approach for the solvent.3 Other hybridmodels combine the MD approach for the polymers with themulti-particle collision dynamics (MPCD) or Stochastic Rota-tion Dynamics (SRD) method for the solvent.4–10 Mesoscopicmethods like Dissipative Particle Dynamics (DPD)11–13 havealso been employed to study polymer solutions by represent-ing the solvent with DPD particles acting as “fluid elements”and representing the polymer molecules by a chain of suchDPD particles connected to each other with springs.14–22 Morerecently, a technique called Smoothed Dissipative ParticleDynamics (SDPD) has been developed,23,24 which is a modi-fied version of Smoothed Particle Hydrodynamics (SPH)25–29

with added DPD-like pairwise thermal fluctuation terms. Thistechnique has also been used for simulating polymer solutionsin a similar way like DPD.30–32

Furthermore, there are other simulation techniques ona much higher level of coarse-graining based on Browniandynamics such as Responsive Particle Dynamics (RaPiD),33,34

in which the solvent is not explicitly modeled but is ratherimplicitly present and the polymer molecules are representedby their centers-of-mass and interact through pair-potentials.RaPiD has been used for studying various phenomena relatedto polymer solutions.35–37 Recently, another technique basedon Brownian dynamics with an implicit background fluid hav-ing hydrodynamic interactions was developed for modelingself-developing flow of polymer solutions in the bulk as well asin the presence of solid interfaces.38,39 As the background fluidis implicitly present at the positions of the polymer molecules,this technique is computationally efficient as it does not needseparate position updates for the fluid particles. Nevertheless,the resolution of the background fluid is fixed by the concentra-tion of the polymers and one is not free to choose an arbitraryequation of state for the fluid. To resolve these problems, itwas envisaged that two different particles be used—one formodeling polymers and the other for the fluid. However, thiscan only be done if a proper interaction term is constructedthat not only couples the motion of the polymers and the fluidbut also conserves momentum so as to preserve long rangehydrodynamics. This is precisely what we present in this paper.Furthermore, we have derived pairwise fluctuation terms forthe fluid using the Fokker-Planck equation so that the steadystate probability distributions of the positions and velocitiesof the fluid in a quiescent state correspond to the equilibriumdistribution. We arrive essentially at the same result that hasbeen derived using General Equation for the Non-EquilibriumReversible-Irreversible Coupling (GENERIC) for an incom-pressible SDPD fluid.23,24 However, these fluctuation termsmay be neglected if they do not play a major role in the

case being studied, under which circumstance the fluid modelreduces to standard SPH.

II. MODEL DEVELOPMENTA. Equation of motion for the polymer blobs

Consider the motion of a mesoscopic polymer blob a dis-solved in a solvent, alternatively referred to as the backgroundfluid in this work, under flowing conditions. It is well knownthat for overdamped systems, the time scale over which theparticles move to any significant extent is orders of magni-tude larger than the time scale over which their velocities getcompletely thermalized. Therefore, these velocities may beintegrated out and the positions of the particles can be updatedusing a first order Brownian dynamics propagator,40,41

dra = v(ra)dt +

(Fa

ξa

)dt + kBT

∂

∂ra

(1ξa

)dt + dWr

a. (1)

We would like to emphasize that v(ra), shorthand for v(ra, t),is not the velocity of the polymer blob a but rather the back-ground fluid velocity at the position of polymer blob a at timet. Fa, shorthand for Fa(t), is the driving force acting on poly-mer blob a as a result of the interaction with other polymerblobs, in addition to any force field that may have been applied.In the remaining part of the paper, we will not include t inour notation, tacitly assuming that it is implicitly present. ξa,shorthand for ξ(ra), is the friction coefficient at the positionof polymer blob a. The third term on the right-hand side ofEq. (1) is a drift term accounting for the spatial variation ofthe friction coefficient, which we have neglected in this studyas we have assumed a constant friction coefficient for the sakeof simplicity. The last term on the right-hand side of Eq. (1),i.e., dWr

a, is a random displacement that is uncorrelated in timeand has a magnitude that is calculated in accordance with thefluctuation dissipation theorem, satisfying

〈dWradWr

b〉 = 2kBT

(dtξa

)δabI. (2)

Since the background flow field is typically not knowna priori, it must be calculated during the simulation in real time.We calculate this background velocity field on a moving gridof fluid blobs which serve as the node points as in a SmoothedParticle Hydrodynamics (SPH) simulation.25–29 Since the flowfield is calculated only at a finite number of node points, thebackground fluid velocity at the position of the polymer bloba in Eq. (1) must be estimated based on interpolation. To thisend, we use the integral interpolant

v(ra) =∫w f (|r − ra |)v(r)d3r, (3)

where d3r is a volume element and w f (r) is a normalizedweight function with a cutoff Rc satisfying for a 3-D simula-tion: ∫

Rc0 4πr2w f (r)dr = 1. The superscript f inw f (r) indicates

that this is the weight function used for the fluid blobs. Theintegral in Eq. (3) can be replaced by a summation runningover each of the fluid neighbors i of polymer blob a using thestandard SPH formulation

034902-3 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

v(ra) =Nf∑i=1

w f (rai)

nfi

vi. (4)

Here, N f is the total number of fluid blobs and rai, shorthandfor |ra � ri |, is the distance between polymer blob a and fluidblob i at time t. nf

i is the local number density of fluid blobsat the position of the fluid blob i at time t, which is calculatedby running over each of the fluid neighbors j of the fluid blobi using the aforementioned weight function w f (r),

nfi =

Nf∑j=1

w f (rij). (5)

For calculating the driving force Fa in the second term ofEq. (1), we resort to the Brownian dynamics based polymermodel RaPiD,33 in which the driving force is calculated as thegradient of a two-part pair-potential,

Fa = −∂

∂ra[Φc + Φt], (6)

where Φc is the conservative potential and Φt is the transientpotential, both of which shall be defined and described inSec. III.

It is interesting to note that by rearranging Eq. (1), usingEq. (5), and assuming a constant friction coefficient ξ for thesake of simplicity, one may obtain the effective frictional forceFpf

a acting on the polymer blob a at time t, opposing its motionrelative to the background flow field, as

Fpfa = −ξ

Nf∑i=1

w f (rai)

nfi

(dr∗adt− vi

). (7)

Here, dr∗a/dt is the deterministic part of the rate of change ofthe position of the polymer blob a, i.e., as calculated withoutthe random fluctuation dWr

a and vi is the velocity of the fluidblob i at time t. Furthermore, it can be said that it is throughthis frictional force that we essentially couple the motion ofthe polymer blobs with the motion of the background fluid.In Subsection II B, we shall refer to this frictional force tomotivate the interaction force acting on the fluid blobs due tothe polymer blobs.

B. Equation of motion for the fluid blobs

For calculating the background flow field, we use theincompressible Navier-Stokes equation for fluctuating hydro-dynamics,42–45 augmented with an additional interaction termrepresenting the force acting on the fluid due to the polymer,

DvDt

(r) = −

(∇Pρ

)(r) + η

(∇2vρ

)(r) + g(r)+Ffp(r) + dWv(r).

(8)

Here, the term on the left-hand side is the material derivativeof the velocity v(r), ρ is the mass density of the fluid, η is theviscosity of the fluid,∇P(r) is the pressure gradient, g(r) is theacceleration due to body forces, Ffp(r) is the interaction termwhich represents the force on the fluid due to the polymer, anddWv(r) is a random fluctuation term.

Now, instead of solving the above equation on an Euleriangrid as in CFD, we discretize the above equation on a movinggrid of fluid blobs as in SPH.25–29 Since the fluid blobs them-selves move with the flow field, the position of a fluid blob ican be updated using

dri = vidt. (9)

Here, once again we clarify that ri is the position of fluidblob i at time t and vi is the velocity of that fluid blob at thattime. As a direct consequence of this, the material derivativeof the velocity can be readily calculated as the rate at whichthe velocity of the fluid blobs changes in the Lagrangian frameof reference,

DvDt

(ri) =dvi

dt, (10)

and g(r) at the position of fluid blob i can be discretized as g(i).For calculating the pressure gradient and the viscous dissipa-tion term, standard SPH finite-difference-like forms29 havebeen employed,(

∇Pρ

)(ri) =

1m

Nf∑j=1

*.,

Pi

(nfi )2

+Pj

(nfj )2

+/-

dw f

dr(rij)

rij

rij, (11)

(∇2vρ

)(ri) =

1m

Nf∑j=1

*.,

2

nfi nf

j

+/-

dw f

dr(rij)

vij

rij, (12)

where vij is the velocity of fluid blob i relative to fluid blob jand m is the mass of the fluid blobs, which is calculated as theratio of the mass density of the fluid ρ to the average numberdensity of the fluid blobs nf , i.e., m = ρ/nf .

Ffp(r) discretized at the position of the fluid blob i isdenoted by Ffp

i , which represents the influence of the surround-ing polymer blobs on the fluid blob i at time t. We may nowrefer to the frictional force Fpf

a shown in Eq. (7) and constructthe interaction term Ffp

i such that for each individual polymer-fluid pair of polymer blob a and fluid blob i, the force exertedby polymer blob a on fluid blob i is equal and opposite tothe frictional force exerted by fluid blob i on polymer blob a.Thus, the total force on fluid blob i, calculated by summingover the forces due to all the individual polymer blobs, wouldthen be

Ffpi = ξ

Np∑a=1

w f (rai)

nfi

(dr∗adt− vi

), (13)

where Np is the total number of polymer blobs. However, dueto computational reasons, which shall be explained at the endof this subsection, we adopt a slightly modified version for theinteraction term

Ffpi = ξ

Np∑a=1

w f (rai)

nfi

(dr∗adt− v(ra)

), (14)

where v(ra) is the velocity of the fluid interpolated at the posi-tion of polymer blob a at time t, as defined in Subsection II A inEq. (4). Although this means that we do not achieve pairwisemomentum conservation for the polymer-fluid interactions, westill achieve momentum conservation on a local level in such away that the total frictional force exerted on any given polymerblob a by the fluid is distributed back in its entirety to the fluid

034902-4 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

blobs in its local vicinity, the proof of which has been providedin Appendix A. We can further simplify Ffp

i using Eqs. (1) and(14) to arrive at

Ffpi =

Np∑a=1

w f (rai)

nfi

Fa. (15)

We must emphasize that it is through this term that the fluidblobs experience the influence of the polymer blobs. Thus, thetwo interaction terms Fpf and Ffp essentially make this modela two-way coupling scheme. We might add that this strategy ofredistributing the force is reminiscent of a two-way couplingmodel that was developed for gas-fluidized beds.46

Combining the results from Eqs. (8), (10)–(12), and (15)and discretizing the fluctuation term dWv(r) at the position ofthe fluid blob i as a pairwise sum of random terms denotedby dWv

ij, we arrive at the following equation that we haveused in our simulations to update the velocities of the fluidblobs:

dvi = −dtm

Nf∑j=1

*.,

Pi

(nfi )2

+Pj

(nfj )2

+/-

dw f

dr(rij)

rij

rij+

Nf∑j=1

fijvij

+ gidt +dtm

Np∑a=1

w f (rai)

nfi

Fa

+

Nf∑j=1

dWvij, (16)

where f ij, shorthand for f (rij), is a symmetric function definedas follows:

f (rij) =

−

(2η

nfi nf

j

)1rij

dwf

dr (rij) for rij ≤ Rc,

0 for rij ≥ Rc.

(17)

The pairwise velocity fluctuation terms are uncorrelated intime and have been calculated in an anti-symmetric mannersuch that dWv

ij = −dWvji so that the velocity updates are

momentum conserving. The properties of these momentumfluctuations dWij have been calculated in a way that the steadystate probability distribution of the positions and velocities ofthe fluid in a quiescent state yields the expected equilibriumdistribution. From a detailed derivation shown in Appendix B,we have⟨

dWvijdWv

ij

⟩=

(2kBT

m

) (dtm

)fijI, (18)⟨

dWvikdWv

jl

⟩= 0 (ik , jl ∧ ik , lj). (19)

Notice that if instead of Eq. (14) a pairwise anti-symmetricinteraction term as shown in Eq. (13) would have been used,then we would have ended up with an additional term pro-portional to the product of the time step dt and the frictioncoefficient ξ in the equation of motion for the fluid blobs. Thiswould deprive us of the whole advantage of using Browniandynamics for the polymers which allows us to use a larger timestep for overdamped systems where the friction coefficient ξ islarge. This is the reason for not using Eq. (13) as the interactionterm for the fluid blobs.

It is also interesting to note that if we would not haveneglected the spatial variation of the friction coefficient inEq. (1), then we would have had an additional term propor-tional to ∂

∂raln(ξa) in the equation of motion for the fluid blobs.

However, since this term grows only weakly with increasingξa, it would not affect our proposed technique significantly.

III. TEST SYSTEM AND PARAMETERSA. The conservative potential

We have tested our technique with a model linear polymersolution at different concentrations. We use the Flory-Hugginspotential, which has been used in the literature for describingthe conservative part of the interaction potential between thelinear polymers,35–37 defined as follows:

Φc = pkBTNp∑

a=1

[(1 − φa

φa

)ln(1 − φa) − χφa

]. (20)

Here, Np is the total number of polymer blobs in solution asmentioned before, Φc is the conservative potential, p is thenumber of Kuhn segments in a polymer blob, χ is the solventinteraction parameter, and φa is the local volume fraction ofpolymer blobs in the neighborhood of polymer blob a at timet calculated as

φa =np

a

npmax

. (21)

Here, npmax is the maximum number density of polymers that

the system is allowed to reach, i.e., the melt density, and npa is

the local number density of polymer blobs at the position ofpolymer blob a calculated as

npa =

Np∑b=1

wp(rab), (22)

where rab is the distance between polymers a and b at timet and wp(r) is a normalized weight function with a cutoff rc

satisfying for a 3-D simulation: ∫rc

0 4πr2wp(r)dr = 1. Thesuperscript p in wp(r) indicates that this is the weight functionused for the polymer blobs.

B. The transient potential

We have used the RaPiD technique to incorporate memoryeffects into the simulation model through a transient potential,which depends on the history of the interacting polymers bykeeping track of dynamic variables, given by33,34

Φt =12α

Np∑a,b=1

(λab − λ

eqab

)2. (23)

Here, Φt is the transient potential, α is a parameter associatedwith the strength of the interactions or, in other words, thepenalty for the deviation of the dynamic variable λab from itsequilibrium value λeq

ab. The variable λab, shorthand for λab(t),is a dimensionless dynamic variable representing the degreeof intermixing of the polymers a and b, which evolves overtime based on the following first order stochastic differentialequation:

dλab = −(λab − λeqab)

dtτ

+ dWλab, (24)

where τ is the relaxation time and Wλab is a Wiener process

with time-uncorrelated increments satisfying

034902-5 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

〈dWλabdWλ

ab〉 =

(2kBTα

) (dtτ

)I. (25)

The variable λeqab, shorthand for λeq(rab), is the equilibrium

value of the variable λab for the distance rab, shorthand forrab(t), between polymers a and b at time t. For the definitionof λeq

ab, we use here the following form that has been used inthe literature:35–37

λeq(rab) =

(1 − rab

rc

)2for rab ≤ rc,

0 for rab ≥ rc.(26)

C. The equation of state

For the fluid, we use the commonly used pseudo-incompressible equation of state, which is essentially a mod-ified version of the Tait equation of state with the exponentchosen as 7 so that it works well for water,47,48

Pi = P0

*,

nfi

nf+-

7

− 1

, (27)

where P0 is chosen such that the velocity of sound in the simu-lation is sufficiently large in order that the density fluctuationsare sufficiently small, which results in a fluid that resemblesan incompressible fluid.

D. Definition of weight functionsand system parameters

For the polymer blobs, we have used a normalized weightfunction that has been used earlier in the literature, where thepolymer model RaPiD has been employed.35–37 It is a mono-tonically decreasing function with derivatives up to the firstorder continuous to prevent any discontinuities in the forceswhich are related to the first derivative of the weight func-tion. Furthermore, it has a non-zero derivative at the origin inorder to have a net repulsive force at zero-distance to preventclustering. It is given by wp(r),

wp(r) =

152π(r5

c−σ5)

(rc − σ)(rc + σ − 2r) for r ≤ σ,

152π(r5

c−σ5)

(r − rc)2 for σ ≤ r ≤ rc,

0 for r ≥ rc,

(28)

where the cutoff rc is chosen as 2.5σ. It can be easily checkedthat the weight function wp(r) is normalized in 3-dimensionswithin this cutoff radius rc such that ∫

rc0 4πr2wp(r)dr = 1.

For the fluid blobs, we have chosen the normalized M4

kernel commonly used in SPH29 as the weight function. It isa cubic spline (piecewise continuous polynomial of degree 3)having derivatives up to the second order continuous and acutoff of Rc = 2h. It is given by w f (r),

w f (r) =

14πh3

[(2 − r

h )3 − 4(1 − rh )3

]for r ≤ h,

14πh3 (2 − r

h )3 for h ≤ r ≤ 2h,

0 for r ≥ 2h,

(29)

where h is what is commonly referred to as the support ofthe weight function. Again, it can be easily checked that theweight function w f (r) is normalized in 3-dimensions withinthe cutoff radius such that ∫

Rc0 4πr2w f (r)dr = 1.

We have chosen h = 2σ for our simulations such that thecutoff radius Rc for the fluid blobs is larger than the cutoffradius rc chosen for the polymer blobs because the weightfunction w f (r) must be able to accurately estimate the second-order derivatives occurring in the equation of motion for thefluid blobs. Following the same logic, it immediately followsthat for the polymer-fluid interactions, as we do not need tocalculate any gradients, we can use the weight function wp(r)with a smaller cutoff rc instead of w f (r) with a larger cutoff Rc

for computational efficiency. The values of the other systemparameters have been summarized in Table I.

The physical properties of the fluid, i.e., the density andviscosity, have been chosen to be consistent with those ofwater. The value of P0 has been chosen high enough to ensuresmall density fluctuations and a large enough velocity of soundcs in the simulation, which can be calculated as

cs =

√∂P∂ρ=

√7P0

ρ. (30)

For the value of P0 that we have used, the velocity of soundin the simulation calculated using the above equation is about0.03 m/s. We have ensured that the velocities that we encounterin the simulations that we have performed in this study aremuch smaller than this velocity of sound in our simulation.

The length scale σ has been purposefully chosen to belarge for computational reasons so as to be able to use a largetime step as the main intention here is to show the two-waycoupling between the polymers and the fluid through our inter-action term. Although for the fluid blobs this length scale isreasonable as it is motivated phenomenologically using SPH,for the polymer blobs which are motivated using a polymermodel at the mesoscopic level, it is much larger than the usualmolecular level. However, if one were to use polymers whichare much smaller, then one must also use smaller fluid blobsbecause otherwise the velocity gradients of the fluid will not be

TABLE I. Summary of system parameters.

System parameter Symbol Value Unit

Length scale σ 5.0 µmTime step dt 1.0 µsTemperature T 300 KDensity of fluid ρ 1000 kg/m3

Viscosity of fluid η 1.0 mPa sResolution of fluid nf 1.9099 × 1015 Particles (m3)Pressure coefficient P0 0.13 PaFriction coefficient ξ 1.0 × 10�7 kg/sStrength of polymer interactions α 500 kBTRelaxation time τ 1.0 sNumber of Kuhn segments p 300 000 . . .

Maximum number density npmax 1.0 × 104 C∗

of polymersFlory-Huggins interaction χ 0.5 . . .

parameter

034902-6 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

accurately experienced by the polymer blobs thereby adverselyaffecting the hydrodynamic coupling. This would lead to asmaller time step and, on the other hand, if larger fluid blobsare used, then the number of polymers required becomes verylarge and we run into computational problems again as willbe described in more detail in Sec. V. Furthermore, due to thelarge size of the polymer blobs, the concentration C∗, whichis calculated as the concentration at which there is on aver-age 1 particle in a volume of a sphere of radius σ, is rathersmall because this sphere is quite large. Hence, we have chosenlarge values for parameters like the effective Kuhn segments pof our polymer blob, the strength of inter-polymer interactionsα, and the maximum number density of polymers np

max, whichis a Flory-Huggins parameter that typically corresponds to themelt density. For the Flory-Huggins interaction parameter χ,we have chosen the limiting value of 0.5, above which phaseseparation would occur, which is not what we aim for in thisstudy. Nevertheless, depending on the system being modeled,i.e., depending on whether the solvent is a good solvent or abad solvent for the polymer, the value of the Flory-Hugginsparameter χ may be selected accordingly and our method ofcoupling the polymer and the fluid would still work. We wantto emphasize that the term polymer blob in this paper refersto a whole polymer molecule or even a collection of a fewpolymer molecules in a single blob, which is slightly differentfrom the usual usage of the word blob referring to a collectionof a few monomers.

IV. RESULTS AND DISCUSSIONA. Pure fluid simulations

In this subsection, we present simulations of the purefluid, i.e., in the absence of the polymers. Since the equationof motion for the fluid is based on the Navier-Stokes equa-tion for an incompressible Newtonian fluid, we expect to seethe Newtonian behavior in the dynamic properties of the purefluid.

1. Equilibrium simulation

Consider a fluid blob in a sea of fluid blobs of an incom-pressible Newtonian fluid in a quiescent state. At equilibrium,although the mean velocity of the fluid blob is zero, at anygiven instant it has a finite instantaneous velocity due to ther-mal fluctuations. Although these instantaneous velocities arerandom both in magnitude and direction, we can still say some-thing about the distribution of these velocities and the rate atwhich these velocities dissipate from the kinetic and hydrody-namic theories, respectively. It is well known that at thermalequilibrium, the velocities must obey the Maxwell-Boltzmanndistribution, and therefore for any given component of velocityv0, we have

〈v20 〉 = kBT/m, (31)

where kB is the Boltzmann constant, T is the temperature ofthe system, and m is the mass of the fluid blob. Furthermore,from hydrodynamic theory for Newtonian liquids, it is knownthat at short times and large wavelengths, the transverse cur-rent correlation function decays exponentially over time andthe time constant is related to the kinematic shear viscosity

FIG. 1. Normalized transverse current correlation function for the pure fluidat equilibrium for three different wave vectors.

of the system. So, we have calculated the transverse currentcorrelation function Ct(k, t) for velocities in the y-directionand wave vectors in the x-direction (kx = kex),

Ct(k, t) =1

Nf〈jy(kx, t)j∗y(kx, 0)〉, (32)

where N f is the total number of fluid blobs and the transversecurrent jy(kx, t) is defined as follows:

jy(kx, t) = ey.Nf∑j=1

vj(t)e−ikx .rj(t), (33)

where rj(t) is the position of fluid blob j at time t and vj(t)is its instantaneous velocity at that time. From hydrodynamictheory, it is known that at short times and small k-values, thetransverse current correlation function Ct(k, t), defined above,

FIG. 2. Flow curve for the pure fluid model with and without thermalfluctuations.

FIG. 3. Reverse Poiseuille flow profile—comparison of simulation resultswith theory.

034902-7 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

FIG. 4. Flow curves for different con-centrations of polymer solution showingsplit-up of the various contributions tothe shear viscosity. (a) Concentration C1= 2.5 C∗. (b) Concentration C2 = 5 C∗.

decays as follows:49

Ct(k, t) = 〈v20 〉e−νk2t , (34)

where ν is the kinematic shear viscosity of the fluid and v0 isthe initial value of the y-component of the velocity. CombiningEqs. (31) and (34), we have

Ct(k, t) =

(kBTm

)e−νk2t . (35)

Since we need small k-values, i.e., large wavelengths in thex-direction, we have chosen a simulation box with a lengthLx = 33.75σ in the x-direction and 10σ in the other two-directions and three different k values of 2π/Lx, 4π/Lx, and6π/Lx, i.e., 0.1862σ�1, 0.3723σ�1, and 0.5585σ�1, respec-tively. As we can see from Fig. 1, there is a good matchbetween the simulation results and the theoretical curves, par-ticularly for the smallest k-value at small times, as was to beexpected, indicating that the model is hydrodynamically con-sistent. Furthermore, it ascertains that the velocity fluctuationterms that we have derived based on the Fokker-Planck equa-tion in Appendix B do indeed maintain the correct kinetictemperature and that the system has the right kinematic shearviscosity.

2. Shear flow simulations

We have performed shear flow simulations with our fluidmodel for a number of different shear rates. Essentially, inevery simulation, we have maintained a constant shear rateacross the box using the Lees-Edwards technique50 and mea-sured the average stress in the box over a period of time. Fromthis, we have calculated how the viscosity varies with the shearrate as shown in Fig. 2.

We have performed these shear flow simulations usingour fluid model with thermal fluctuations as well as by turningoff the fluctuations. As we can see from Fig. 2, the viscosityis close to the viscosity of water (1 mPa s) for the range ofshear rates that we have considered, given the parameters andresolution of the fluid that we have used. Although the viscos-ity does vary by about 15%, this is to be expected given thefinite resolution of the fluid typically used in SPH simulations.More importantly, it is consistent with the viscosity that canbe interpreted from the transverse current correlation functionshown in Subsection IV A 1.

3. Reverse Poiseuille flow simulation

In this subsection, we present simulations where the shearrate varies across the box in the y-direction as opposed to the

simulations shown in Subsection IV A 2, where there was ahomogeneous shear flow. We essentially achieve this by apply-ing a constant gravitational force field in the x-direction in thetop half of the box and an equal and opposite field in the lowerhalf of the box which leads to the development of a Poiseuilleflow profile in both halves of the box. This method is what iscommonly referred to as the “Reverse Poiseuille flow” in theliterature.51 We have performed this simulation for the fluidmodel with and without fluctuations and compared the resultswith the analytical profile that can be calculated from a theorybased on the Navier-Stokes equation for a Newtonian fluid.As can be seen from Fig. 3, there is a good match betweensimulations and theory.

Furthermore, it can be seen from Figs. 2 and 3 that thefluctuation terms do not affect the fluid viscosity. Hence, forthe sake of computational efficiency, we do not include thethermal fluctuations in Sec. IV B where we have polymer blobsas well.

B. Polymer solution simulations

For this study, we have considered an aqueous solu-tion of a model polymer system, where the polymer blobsinteract with each other through the conservative and tran-sient interactions formulated by the RaPiD model mentionedin Subsections III A and III B. The model for the aqueoussolvent is the same as the one we have used in Subsec-tion IV A but without thermal fluctuations. The polymersand the fluid blobs of the aqueous solvent interact with eachother through the interaction terms that we have defined inSec. II.

1. Shear flow simulations

We have performed shear flow simulations for our modellinear polymer solutions of two different concentrationsC1 = 2.5 C∗ and C2 = 5 C∗ and the results have been plot-ted in Fig. 4. The different contributions to the shear viscositydue to the polymers, the fluid, and the interaction term have

TABLE II. Parameters for the Carreau model fit.

Fit parameter Symbol C1 C2 Unit

Zero shear rate viscosity η0 2.5260 6.6688 mPa sInfinite shear rate viscosity η∞ 0.9 0.9 mPa sFit parameter 1 λ 3.0504 2.1408 sFit parameter 2 n 0.3441 0.0008 . . .

034902-8 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

FIG. 5. Reverse Poiseuille flow profilesfor different concentrations of polymersolutions showing the comparison of thepolymer and fluid velocity profiles in thesimulation with the CFD profile gener-ated using COMSOL and the analyticalNewtonian profile. (a) Concentration C1= 2.5 C∗. (b) Concentration C2 = 5 C∗.

been shown. We have fitted the total shear viscosity flow curvewith the Carreau model, which is commonly used for polymersolutions, given by52

η − η∞η0 − η∞

= [1 + (λγ)2]n−1

2 , (36)

where η0 is the zero shear rate viscosity, η∞ is the infinite shearrate viscosity, λ is a parameter with units of time, and n is adimensionless parameter.

The fit parameters of the Carreau model that best repre-sents the total viscosity for the two different concentrations ofpolymer solutions that we have studied are presented in TableII. As can be seen from Fig. 4, the solution is non-Newtonianin nature and the effect is clearly significant at higher con-centrations. This shear thinning is mainly due to the transientinteractions between the polymer blobs which are weaker athigher shear rates.

2. Reverse Poiseuille flow simulations

We have also performed reverse Poiseuille flow simu-lations for our model linear polymer solutions at the twoconcentrations C1 = 2.5 C∗ and C2 = 5 C∗. We have comparedthe results from our simulations with CFD simulations per-formed with the commercial package COMSOL53 to whichwe fed as input the Carreau model flow curve that we haveshown in Subsection IV B 1. The results have been plotted inFig. 5.

FIG. 6. Normalized density profile for fluid and polymer blobs for differentconcentrations of polymer solutions, viz., C1 = 2.5 C∗ and C2 = 5 C∗. Notethat the density of the blobs has been normalized with the mean density ofthose blobs calculated with the respective weight function in that particularsimulation.

As can be seen from Fig. 5, there is no lag between thepolymer blobs and the fluid blobs as their velocity profilescoincide with each other, which indicates that our interactionterm couples the fluid and the polymers well. Furthermore,our model for the non-Newtonian polymer solutions is able toreproduce the characteristic flattening of the velocity profile, asis evident by comparison with the parabolic Newtonian profileparticularly for higher concentrations.

Our simulations also show the cross-stream migrationphenomenon observed for polymers in solution.54 As can beseen from Fig. 6, the concentration of water blobs remains con-stant along the y-axis, i.e., along the gradient direction. Thepolymer concentration, however, has minima at the regionsof maximum shear and maxima at the regions of zero shear.Notice that the densities of the blobs have been normalizedwith their respective mean density in that particular solution.Therefore, the maximum concentration of the polymers in thezero-shear region in absolute terms is higher for the poly-mer solution with concentration C2 as compared to that forthe polymer solution with concentration C1 although it mightseem otherwise on a first glance by looking at the normalizeddensities plotted in Fig. 6.

V. CONCLUSION AND SCOPEFOR FURTHER RESEARCH

We have presented a novel technique, hereafter referred toas Hydrodynamically Coupled Brownian Dynamics (HCBD),which is essentially a momentum conserving two-way cou-pling algorithm that couples the motion of polymers movingas per Brownian dynamics with the motion of the fluid whichis calculated using the SPH methodology based on the numer-ical solution of the Navier-Stokes equation discretized on amoving grid of fluid particles. We have calculated fluctuationterms for the update of the fluid velocities based on the Fokker-Planck equation, which effectively renders our fluid modelinto a discretized version of fluctuating hydrodynamics.42–45

We essentially arrive at the result for an incompressible SDPDfluid derived using the GENERIC approach.23,24 Nevertheless,for computational efficiency, these fluctuating terms may bedropped out and then the pure fluid model, i.e., in the absenceof the polymers, reduces to the standard SPH formulation.The main achievement of this study is the construction of theinteraction term between the Brownian polymer model andthe SPH fluid model. This interaction term couples the motionof the polymers to that of the fluid without any lag and also

034902-9 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

conserves momentum at a local level, preserving long rangehydrodynamics.

The merits of this HCBD model as a technique for mod-eling self-developing flow of polymer solutions can be betterunderstood if it is compared with a standard Brownian dynam-ics simulation. In a standard Brownian dynamics simulation,if the friction is to be applied with respect to a non-static back-ground fluid, then an additional term describing the motion ofthe background fluid must be added to the Brownian dynamicspropagator. However, typically the background fluid velocityis not known a priori particularly if the fluid is flowing througha complex geometry. Even if independent simulations are car-ried out to determine the flow profile of the fluid, which then isadded to the motion of the Brownian propagator for the poly-mers, there is still a lag observed between the polymers andthe fluid. More importantly, the fluid velocity profile experi-ences no influence from the motion of the polymers, therebymaking it a one-way coupling scheme. In our HCBD scheme,on the other hand, the fluid and the polymer are coupled witheach other through the frictional interaction term that doesnot lead to any lag and makes it a two-way coupling scheme.This enables us to model the flow of the polymer solution asit develops in space and time. Furthermore, since our fluidmodel is based on SPH, it is easy to incorporate boundaryconditions such as the no-slip boundary condition at the solidinterfaces in a similar way to what we have shown in an earlierwork.39

One of the consequences of modeling the polymers ata mesoscopic scale while describing the fluid using a phe-nomenological approach is that if one wishes to accuratelymodel the scale of the polymers, then one must also use fluidblobs that are not orders of magnitude larger, otherwise veloc-ity gradients of the fluid are no longer captured accurately.This can affect the hydrodynamics of the system adversely.Even so, if larger fluid blobs are used so that a larger time stepmay be used, still the ratio between the number of polymersand the number of fluid blobs becomes large, making the sim-ulation computationally intensive. However, the computationtime may be substantially reduced by massive parallelization,which in principle, our propagators allow for and it would stillbe faster than an MD simulation of course as we exploit theadvantage of coarse-graining.

It is also interesting to note that if one imagines a hypothet-ical situation in which a fluid blob is centered at the position ofthe center-of-mass of each polymer blob, not necessarily thesame fluid blob at all times, then the position update for thepolymer blob and the velocity update for the fluid blob more-or-less reduce to that of the model described in our recentlypublished work, in which the fluid velocities are calculated atthe positions of the centers-of-mass of the polymers and henceno interpolation is required.38,39 In that model, it was proposedthat the force on the polymers be immediately transmitted tothe fluid, the rationality of which, in a way, has been justifiedthrough this paper, where we indeed arrive at it through theconstruction of the frictional force between the polymers andthe fluid which couples the two together. Having said that, themodel presented in this present paper has several advantagesover the earlier model such as being able to independentlychoose an equation of state for the fluid through the pressure

term and being able to independently decide the resolution offluid irrespective of the concentration of the polymers. More-over, although it might seem that the earlier model is moreefficient than the present model because the former modeldid not have to move the fluid around separately, yet the lat-ter model can actually turn out to be computationally moreefficient in certain cases. This is because we can reduce thenumber of fluid particles in the system by choosing a resolutionindependent of the concentration of the polymers limited onlyby the dimensions of the complex geometry through which itflows.

Another important point is that the methodology pre-sented in this paper allows one to use any polymer modelof one’s choice and combine it with the SPH fluid model. Thepolymer model that we have used in this study, i.e., RaPiD, hasthe limitation that it does not effectively capture all the internaldynamics of the polymer molecules but rather mainly relieson inter-polymer interactions to produce a non-Newtonianresponse like the shear-thinning behavior. However, as men-tioned before, using our technique of coupling presented inthis paper, i.e., HCBD, one can couple the SPH-based fluidmodel to any other polymer model. For instance, one coulduse an extension of the RaPiD model,37 where the polymeris not just represented by its center-of-mass alone but by afinitely extensible non-linear elastic dumbbell that interactswith other such dumbbells using the usual RaPiD potentials.This would be one way of incorporating some internal dynam-ics into the polymer model thereby more accurately modelingviscoelastic polymeric liquids without affecting the compu-tational efficiency too much. This provides scope for furtherresearch on this topic.

ACKNOWLEDGMENTS

This work is part of the Industrial Partnership Programme(IPP) “Computational sciences for energy research” of theFoundation for Fundamental Research on Matter (FOM),which is part of the Netherlands Organisation for ScientificResearch (NWO). This research programme is co-financed byShell Global Solutions International B.V. J. van der Guchtacknowledges the European Research Council for financialsupport (ERC Consolidator grant Softbreak).

APPENDIX A: PROOF OF LOCAL MOMENTUMCONSERVATION FOR THE INTERACTIONBETWEEN POLYMER AND FLUID BLOBS

Using Eqs. (4) and (14), we obtain the total force due topolymer blob a on the fluid in its local vicinity, denoted by Ffp

a ,as follows:

Ffpa = ξ

Nf∑i=1

w f (rai)

nfi

*.,

dr∗adt−

Nf∑j=1

w f (raj)

nfj

vj+/-

. (A1)

Simplifying the first term using Eq. (5) and rewriting the sec-ond term as two terms, one where j = i and one where j , i,we obtain

034902-10 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

Ffpa = ξ

dr∗adt−

Nf∑i=1

*,

w f (rai)

nfi

+-

2

vi

−*.,

Nf∑i=1

w f (rai)

nfi

Nf∑j,i

w f (raj)

nfj

vj+/-

. (A2)

Rewriting the last term and simplifying, we get

Ffpa = ξ

dr∗adt−

Nf∑i=1

*,

w f (rai)

nfi

+-

2

vi

−*.,

Nf∑j=1

w f (raj)

nfj

vj

Nf∑i,j

w f (rai)

nfi

+/-

. (A3)

Using Eq. (5) and rewriting the last term, we obtain

Ffpa = ξ

dr∗adt−

Nf∑i=1

*,

w f (rai)

nfi

+-

2

vi

−

Nf∑j=1

w f (raj)

nfj

vj*.,1 −

w f (raj)

nfj

+/-

. (A4)

Simplifying further and noting that two of the terms canceleach other and replacing j with i, we get

Ffpa = ξ

*.,

dr∗adt−

Nf∑i=1

w f (rai)

nfi

vi+/-

, (A5)

which is clearly equal and opposite to the total force exertedby the fluid on the polymer blob a, i.e., Fpf

a shown inEq. (7).

APPENDIX B: DERIVATION OF STOCHASTICUPDATE FOR THE FLUID VELOCITIES

In this appendix, we derive the statistical properties ofthe velocity fluctuations dWv

jk for the pure fluid at quies-cent state. Although we use a different approach, we essen-tially arrive at the same result as the velocity fluctuations foran incompressible SDPD fluid derived using the GENERICapproach.23,24

According to the Chapman-Kolmogorov equation, theprobability to find the system at time t + dt at the point zin the phase space characterised by positions and velocities,i.e., z = {r1, . . . , rN , v1, . . . , vN }, given that at t = 0 the systemwas at point z0 in the phase space, is given by

G(z; z0; t + dt) =∫

d6N z′G(z; z′; dt)G(z′; z0; t), (B1)

i.e., where z′ is any intermediate point in the phasespace, through which the system could have passed at timet with limdt→0 G(z; z′; dt)= δ(z − z′). Going through theusual derivation, we arrive at the following Fokker-Planckequation:

∂G(z; z0; t)∂t

= −∑

i

∑α

∂

∂ri,α

{(vi,α

)G(z; z0; t)

}−

∑i

∑α

∂

∂vi,α

*.,

∑j

fijm

(vj,α − vi,α) +Fi,α

m+/-G(z; z0; t)

−12

∑j

∑k

∑l

∑β

∂

∂vi,β

⟨dW v

ik,αdW vjl,β

⟩dt

G(z; z0; t)

,

(B2)

where α and β run from 1 to 3 corresponding to the differentCartesian components and F i represents the force due to thepressure gradient term,

Fi

m=

(∇P)i

ρi= ∇

(Pρ

)i+

Pi

ρ2i

(∇ρ)i. (B3)

Since the standard SPH formalism for a gradient of any

property A is given by (∇A)i =N∑

j=1

miAiρi

dwdr (rij)

rij

rij, we get

Fi =

Nf∑j=1

*.,

Pj

(nfj )2

+Pi

(nfi )2

+/-

dw f

dr(rij)

rij

rij, (B4)

which is the form that has been shown in Eq. (11) and usedthereafter in Eq. (16).

At steady state, we expect the left-hand side of Eq. (B2) tobe zero. Furthermore, we expect that for a system in quiescentstate, the equilibrium distribution of z is given by the Maxwell-Boltzmann distribution

Geq(z) ∝ exp

1kBTΦ (r1, . . . , rN ) −

12kBT

∑i

mvi · vi

,

(B5)

where Φ is the total potential energy based on the pressuredistribution of the system given by Φ = PV =

∑Nf

i=1 PiV/Nf ,where N f /V may be replaced by nf , from which the conserva-tive forces can be calculated as Fi = �∂Φ/∂ri. At equilibrium,G(z; z0; t) in Eq. (B2) must be replaced by Geq(z).

We pre-calculate some of the derivatives that will berequired as follows:

∂

∂riGeq(z) = −

1kBT

(∂

∂riΦ

)Geq(z) =

FiGeq(z)kBT

, (B6)

∂

∂viGeq(z) = −

mviGeq(z)kBT

. (B7)

Notice that these imply∑i

∑α

∂

∂ri,αvi,αGeq(z) +

∑i

∑α

∂

∂vi,α

Fi,α

mGeq(z) = 0. (B8)

Therefore, these terms will get eliminated from Eq. (B2) atequilibrium. As a result, all remaining sums between curlybrackets in Eq. (B2) with G(z; z0; t) replaced by Geq(z) mustbe identically equal to zero.

We re-write the correlations of the velocity fluctuations asfollows: ⟨

dW vik,αdW v

jl,β

⟩= 2kBTCvv

ikαjlβdt. (B9)

034902-11 Ahuja, van der Gucht, and Briels J. Chem. Phys. 148, 034902 (2018)

Thus, at equilibrium, we must have∑j

fijm

(vj,α − vi,α

)+

∑j

∑k

∑l

∑β

Cvvikαjlβmvi,β = 0. (B10)

These equations will be identically satisfied if we chooseCvv

ikαjlβ = δαβ[(fik/m2)δijδkl − (fki/m2)δilδjk

].

Thus, in conclusion, to get the equilibrium distribution atsteady state, we must choose the stochastic variables accordingto ⟨

dWvijdWv

ij

⟩=

(2kBT

m

) (dtm

)fijI, (B11)⟨

dWvikdWv

jl

⟩= 0 (ik , jl ∧ ik , lj). (B12)

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